Invariance Principle for the Random Walk Conditioned to Have Few Zeros

Abstract

We consider a nearest neighbor random walk on $\mathbb{Z}$ starting at zero, conditioned to return at zero at time 2n and to have a number z _n of zeros on (0, 2n]. As $n \rightarrow +\infty$, if $z_{n} = o(\sqrt{n})$, we show that the rescaled random walk converges toward the Brownian excursion normalized to have unit duration. This generalizes the classical result for the case z _n  ≡ 1.